Multisummability for generalized power series

TL;DR

This paper introduces multisummability for generalized power series with natural support and proves that the expanded real field structure remains o-minimal, incorporating important functions like Gamma and Zeta.

Contribution

It develops a new multisummability theory for generalized power series and establishes o-minimality of the expanded real field including these series and their multisums.

Findings

Proves o-minimality of the structure expanded by multisums of generalized power series.

Shows the structure includes the Gamma and Zeta functions.

Extends the real field with new summability methods for generalized series.

Abstract

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,\infty)$ and the Zeta function on $(1,\infty)$.